starting from a stationary - chino.k12.ca.us each expression. 62/87,21 62/87,21 62/87,21 62/87,21...
TRANSCRIPT
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 1
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 2
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 3
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 4
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 5
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 6
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 7
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 8
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 9
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 10
4-3 Simplifying Radical Expressions
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
Simplify each expression.
1.
SOLUTION:
2.
SOLUTION:
3.
SOLUTION:
4.
SOLUTION:
5.
SOLUTION:
6.
SOLUTION:
7.
SOLUTION:
8.
SOLUTION:
9.
SOLUTION:
10. MULTIPLE CHOICE Which expression is
equivalent to ?
A
B
C
D
SOLUTION:
The correct choice is D.
Simplify each expression.
11.
SOLUTION:
12.
SOLUTION:
13.
SOLUTION:
14.
SOLUTION:
15.
SOLUTION:
16.
SOLUTION:
Simplify each expression.
17.
SOLUTION:
18.
SOLUTION:
19.
SOLUTION:
20.
SOLUTION:
21.
SOLUTION:
22.
SOLUTION:
23.
SOLUTION:
24.
SOLUTION:
25.
SOLUTION:
26.
SOLUTION:
27.
SOLUTION:
28.
SOLUTION:
29.
SOLUTION:
30.
SOLUTION:
31.
SOLUTION:
32.
SOLUTION:
33.
SOLUTION:
34.
SOLUTION:
35. ROLLER COASTER Starting from a stationary position, the velocity v of a roller coaster in feet per second at the bottom of a hill can be approximated by
, where h is the height of the hill in feet.
a. Simplify the equation. b. Determine the velocity of a roller coaster at the bottom of a 134-foot hill.
SOLUTION: a.
b. To determine the velocity of the roller coaster at the bottom of the hill, substitute 134 for h in the
equation .
The roller coaster will have a velocity of about 92.6 ft/sec at the bottom of a 134-foot hill.
36. CCSS PRECISION When fighting a fire, the velocity v of water being pumped into the air is
modeled by the function , where h
represents the maximum height of the water and g
represents the acceleration due to gravity (32 ft/s2).
a. Solve the function for h. b. The Hollowville Fire Department needs a pump that will propel water 80 feet into the air. Will a pumpadvertised to project water with a velocity of 70 feet per second meet their needs? Explain. c. The Jackson Fire Department must purchase a pump that will propel water 90 feet into the air. Will apump that is advertised to project water with a velocity of 77 feet per second meet the fire department’s need? Explain.
SOLUTION: a.
b. To determine the height of the water, substitute 70
for v in the function .
A pump with a velocity of 70 feet per second will pump water only to a maximum height of 76.6 feet. Therefore, this pump will not meet the fire department’s need. c. To determine the height of the water, substitute 77
for v in the function .
A pump with a velocity of 77 feet per second will pump water to a maximum height of 92.6 feet. Therefore, this pump will meet the fire department’s need.
Simplify each expression.
37.
SOLUTION:
38.
SOLUTION:
39.
SOLUTION:
40.
SOLUTION:
41.
SOLUTION:
42.
SOLUTION:
43.
SOLUTION:
44.
SOLUTION:
45.
SOLUTION:
46.
SOLUTION:
47.
SOLUTION:
48.
SOLUTION:
49. ELECTRICITY The amount of current in amperesI that an appliance uses can be calculated using the
formula , where P is the power in watts and
R is the resistance in ohms. a. Simplify the formula. b. How much current does an appliance use if the power used is 75 watts and the resistance is 5 ohms?
SOLUTION: a.
b. Substitute P = 75 and R = 5 in the equation
.
An appliance uses about 3.9 amps of current if the power used is 75 watts and the resistance is 5 ohms.
50. KINETIC ENERGY The speed v of a ball can be
determined by the equation , where k is the
kinetic energy and m is the mass of the ball. a. Simplify the formula if the mass of the ball is 3 kilograms. b. If the ball is traveling 7 meters per second, what is the kinetic energy of the ball in Joules?
SOLUTION: a. If the mass of the ball is 3 kilograms, substitute m = 3 into the equation.
b. Substitute V = 7 in the equation .
The kinetic energy of the ball is 73.5 Joules.
51. SUBMARINES The greatest distance d in miles ththe lookout can see on a clear day is modeled by the
formula . Determine how high the
submarine would have to raise its periscope to see a ship, if the submarine is the given distances away froa ship.
SOLUTION: First solve the equation for h.
Distance 3 6 9 12 15 Height
= d2
h =
(3)
2 = 6
h =
(6)
2 =
24
h =
(9)
2 =
54
h =
(12)2
= 96
h =
(15)2
= 150
52. CCSS STRUCTURE Explain how to solve
.
SOLUTION: To solve an equation of equal ratios, first find the equal cross products and then solve for the variable.
Use the conjugate of to rationalize the denominator.
So, the solution is .
53. CHALLENGE Simplify each expression.
a.
b.
c.
SOLUTION:
a.
b.
c.
54. REASONING Marge takes a number, subtracts 4, multiplies by 4, takes the square root, and takes the
reciprocal to get . What number did she start
with? Write a formula to describe the process.
SOLUTION: Let x = a number.
55. OPEN ENDED Write two binomials of the form
and . Then find their
product.
SOLUTION:
Two binomials of the form and
are and .
56. CHALLENGE Use the Quotient Property of Square Roots to derive the Quadratic Formula by
solving the quadratic equation ax2 + bx + c = 0.
(Hint: Begin by completing the square.)
SOLUTION:
57. WRITING IN MATH Summarize how to write a radical expression in simplest form.
SOLUTION: No radicals can appear in the denominator of a fraction. So, rationalize the denominator to get rid of the radicand in the denominator. Then check if any of the radicands have perfect square factors other than 1. If so, simplify. For example, simplify the following.
58. Jerry’s electric bill is $23 less than his natural gas bill.The two bills are a total of $109. Which of the following equations can be used to find the amount ofhis natural gas bill? A g + g = 109 B 23 + 2g = 109 C g − 23 = 109 D 2g − 23 = 109
SOLUTION: Let g = Jerry’s natural gas bill. Jerry’s electric bill is $23 less than his natural gas bill, so the electric bill is g – 23. The two bills are a total of $109.
So, the correct choice is D.
59. Solve a2 − 2a + 1 = 25.
F −4, −6 G 4, −6 H −4, 6 J 4, 6
SOLUTION: Solve for a.
The roots are –4 and 6. So, the correct choice is H.
60. The expression is equivalent to which of the following?
A
B
C
D
SOLUTION:
So, the correct choice is C.
61. GRIDDED RESPONSE Miki earns $10 an hour and 10% commission on sales. If Miki worked 38 hours and had a total sales of $1275 last week, how much did she make?
SOLUTION: Miki’s earnings are $10 per hour and 10% commission on sales.
Miki made $507.50 last week.
eSolutions Manual - Powered by Cognero Page 11
4-3 Simplifying Radical Expressions